Question

A cylindrical vessel is divided in two parts by a fixed partition which is perfectly heat conducting. The left side of constant volume contains 0.5 moles of gas with $C_v = 2R$ at temperature of 300 K. The right side contains 4 moles of mixture of gas with $C_v = 1.75R$ at same temperature of 300 K. The piston compresses slowly the right side from volume of $V_0$ to $\frac{V_0}{4}$. The walls and piston are thermally insulated from surroundings. The total change in internal energy of gas is $(1000x)J$. Find $x$ if $R=\frac{25}{3}S.I. unit

Answer 1

7.5

Answer 2

The total change in internal energy of the system is equal to the work done on the system, as the walls and piston are thermally insulated. The work is done only on the right side. Assuming slow compression against a constant external pressure equal to the initial pressure of the gas on the right side ($P_i$), the work done on the right side is: $W_{on, right} = P_{ext} \Delta V_{right}$ Assuming $P_{ext} = P_i = \frac{n_R R T_i}{V_0}$. The change in volume is $\Delta V_{right} = V_0 - \frac{V_0}{4} = \frac{3V_0}{4}$. So, $W_{on, right} = \frac{n_R R T_i}{V_0} \times \frac{3V_0}{4} = \frac{3}{4} n_R R T_i$. Given $n_R = 4$ moles, $R = \frac{25}{3}$ J/mol-K, and $T_i = 300$ K. $W_{on, right} = \frac{3}{4} \times 4 \times \frac{25}{3} \times 300 = 7500$ J. The total change in internal energy is $\Delta U_{total} = 7500$ J. Given $\Delta U_{total} = (1000x)$ J, we have $1000x = 7500$, so $x = 7.5$.